IMG_5286.jpeg

# Chapter 3: Data Representation ## What you will learn * Numeric Data Representation * Text Data Representation * Image Data Representation * Audio Data Representation * Video Data Representation ## 3.1 Numeric Data Representation In computing, numeric data is represented using the **...

# Chapter 3: Data Representation ## What you will learn * Numeric Data Representation * Text Data Representation * Image Data Representation * Audio Data Representation * Video Data Representation ## 3.1 Numeric Data Representation In computing, numeric data is represented using the **binary number system**, which consists of 0s and 1s. The binary system is used because it is easy to implement in electronic devices, where 0 can represent an "off" state and 1 can represent an "on" state. ### 3.1.1 Integer Representation **Integers** are whole numbers without any fractional part. They can be positive, negative, or zero. In computers, integers are typically represented using a fixed number of bits. The number of bits used to represent an integer determines the range of values that can be represented. #### 3.1.1.1 Unsigned Integers **Unsigned integers** are non-negative integers (i.e., zero or positive integers). They are represented using the binary number system, where each bit represents a power of 2. For example, an 8-bit unsigned integer can represent values from 0 to 255. The range of an unsigned integer with $n$ bits is from 0 to $2^n - 1$. **Example:** An 8-bit unsigned integer can represent values from 0 to $2^8 - 1 = 255$. | Bits | Minimum Value | Maximum Value | | :--- | :------------ | :------------ | | 8 | 0 | 255 | | 16 | 0 | 65,535 | | 32 | 0 | 4,294,967,295 | #### 3.1.1.2 Signed Integers **Signed integers** can be positive, negative, or zero. They are represented using a variety of methods, including: * **Sign-magnitude:** The most significant bit (MSB) represents the sign of the integer (0 for positive, 1 for negative), and the remaining bits represent the magnitude of the integer. * **One's complement:** Positive numbers are represented as in the sign-magnitude representation. Negative numbers are represented by inverting all the bits of the corresponding positive number. * **Two's complement:** Positive numbers are represented as in the sign-magnitude representation. Negative numbers are represented by inverting all the bits of the corresponding positive number and adding 1. **Two's complement** is the most common method for representing signed integers because it simplifies arithmetic operations. In two's complement, the range of an integer with $n$ bits is from $-2^{n-1}$ to $2^{n-1} - 1$. **Example:** An 8-bit signed integer can represent values from $-2^7 = -128$ to $2^7 - 1 = 127$. | Bits | Minimum Value | Maximum Value | | :--- | :------------ | :------------ | | 8 | -128 | 127 | | 16 | -32,768 | 32,767 | | 32 | -2,147,483,648 | 2,147,483,647 | **Example:** Represent the decimal number -5 in 8-bit two's complement representation. 1. Represent the positive number 5 in binary: 00000101 2. Invert all the bits: 11111010 3. Add 1: 11111011 Therefore, the 8-bit two's complement representation of -5 is 11111011. ### 3.1.2 Floating-Point Representation **Floating-point numbers** are numbers with a fractional part (e.g. 3.14, -0.001). They are represented in computers using a form of scientific notation. The most common standard for representing floating-point numbers is the IEEE 754 standard. A floating-point number is represented by three components: * **Sign bit:** Indicates whether the number is positive or negative (0 for positive, 1 for negative). * **Exponent:** Represents the power of 2 to which the significand is multiplied. * **Significand (also known as mantissa):** Represents the significant digits of the number. There are two common floating-point formats: * **Single-precision (32 bits):** 1 bit for the sign, 8 bits for the exponent, and 23 bits for the significand. * **Double-precision (64 bits):** 1 bit for the sign, 11 bits for the exponent, and 52 bits for the significand. The value of a floating-point number can be calculated as follows: $(-1)^{sign} \times significand \times 2^{exponent - bias}$ Where the bias is a constant value used to allow representation of both positive and negative exponents. For single-precision, the bias is 127, and for double-precision, it is 1023. **Example:** Represent the number 12.5 in single-precision floating-point format. 1. Convert the number to binary: 1100.1 2. Normalize the binary number: $1.1001 \times 2^3$ 3. Determine the sign bit: 0 (positive number) 4. Calculate the exponent: $3 + 127 = 130$. Convert it to binary: 10000010 5. Determine the significand: 1001 (the digits after the decimal point). Pad it with zeros to make it 23 bits: 10010000000000000000000 6. Combine the sign bit, exponent, and significand: 0 10000010 10010000000000000000000 ## 3.2 Text Data Representation In computing, text data is represented using character encoding schemes. A **character encoding scheme** assigns a unique numeric code to each character in a character set. The most common character encoding schemes are ASCII and Unicode. ### 3.2.1 ASCII **ASCII** (American Standard Code for Information Interchange) is a character encoding scheme that uses 7 bits to represent 128 characters, including uppercase and lowercase letters, numbers, punctuation marks, and control characters. ASCII is a simple and widely used character encoding scheme, but it can only represent characters from the English language. ### 3.2.2 Unicode **Unicode** is a character encoding scheme that can represent characters from all the world's languages. It uses a variable number of bits to represent each character, up to 4 bytes (32 bits). The most common Unicode encoding is UTF-8. **UTF-8** (Unicode Transformation Format - 8-bit) is a variable-width character encoding scheme that uses 1 to 4 bytes to represent each character. It is the dominant character encoding for the World Wide Web. Unicode can represent a much larger range of characters than ASCII, making it suitable for representing text in multiple languages. **Example:** The Unicode code point for the letter "A" is U+0041, which is represented in UTF-8 as the single byte 0x41. The Unicode code point for the Euro sign "€" is U+20AC, which is represented in UTF-8 as the three bytes 0xE2 0x82 0xAC. ## 3.3 Image Data Representation Images are represented in computers as a grid of pixels. Each **pixel** represents a single point in the image and has a color value. The color value of a pixel is typically represented using a combination of red, green, and blue (RGB) color components. ### 3.3.1 Raster Graphics **Raster graphics** (also known as bitmap graphics) are images that are represented as a grid of pixels. The color of each pixel is stored in a file, along with the image's width and height. Common raster graphics formats include: * **BMP** (Bitmap): A simple raster graphics format that stores uncompressed pixel data. * **JPEG** (Joint Photographic Experts Group): A lossy compression format that is widely used for storing photographic images. * **PNG** (Portable Network Graphics): A lossless compression format that is commonly used for storing images with sharp lines and text. * **GIF** (Graphics Interchange Format): A lossless compression format that is commonly used for storing simple animations and images with a limited number of colors. ### 3.3.2 Vector Graphics **Vector graphics** are images that are represented using mathematical equations. Instead of storing the color of each pixel, vector graphics store the shapes, lines, and curves that make up the image. Vector graphics are resolution-independent, meaning that they can be scaled up or down without losing quality. They are commonly used for storing logos, illustrations, and other images that need to be scaled without loss of quality. Common vector graphics formats include: * **SVG** (Scalable Vector Graphics): An XML-based vector graphics format that is widely used for web graphics. * **AI** (Adobe Illustrator): A proprietary vector graphics format used by Adobe Illustrator. * **EPS** (Encapsulated PostScript): A vector graphics format used for printing and publishing. ### 3.3.3 Color Representation Colors can be represented in computers using different color models. The most common color models are: * **RGB** (Red, Green, Blue): A color model that represents colors as a combination of red, green, and blue light. Each color component is typically represented using 8 bits, allowing for 256 different shades of each color. * **CMYK** (Cyan, Magenta, Yellow, Black): A color model that represents colors as a combination of cyan, magenta, yellow, and black inks. It's mostly used in the printing process. * **HSV** (Hue, Saturation, Value): A color model that represents colors in terms of hue (the color's position on the color wheel), saturation (the color's intensity), and value (the color's brightness). ## 3.4 Audio Data Representation Audio data is represented in computers as a sequence of samples, where each sample represents the amplitude of the audio signal at a specific point in time. The **sampling rate** is the number of samples taken per second, measured in Hertz (Hz). The **bit depth** is the number of bits used to represent each sample. ### 3.4.1 Digital Audio **Digital audio** is audio that has been converted into a digital format. The process of converting audio from an analog signal to a digital signal is called **analog-to-digital conversion (ADC)**. The reverse process, converting digital audio back to an analog signal, is called **digital-to-analog conversion (DAC)**. Common audio formats include: * **WAV** (Waveform Audio File Format): An uncompressed audio format that stores raw audio samples. * **MP3** (MPEG Audio Layer 3): A lossy compression format that is widely used for storing music. * **AAC** (Advanced Audio Coding): A lossy compression format that is commonly used for storing music and audio on mobile devices. * **FLAC** (Free Lossless Audio Codec): A lossless compression format that is commonly used for archiving audio. ### 3.4.2 Audio Properties Key properties of digital audio include: * **Sampling Rate:** The number of samples per second (e.g., 44.1 kHz for CD quality audio). * **Bit Depth:** The number of bits used to represent each sample (e.g., 16 bits). * **Channels:** The number of audio channels (e.g., mono, stereo). ### 3.4.3 Audio Compression **Audio compression** is the process of reducing the file size of an audio file. There are two types of audio compression: * **Lossy compression:** Reduces the file size by discarding some of the audio data. This can result in a loss of audio quality, but the file size is significantly reduced. * **Lossless compression:** Reduces the file size without discarding any of the audio data. This results in a smaller file size than uncompressed audio, but the audio quality is the same as the original. ## 3.5 Video Data Representation Video data is represented in computers as a sequence of frames, where each frame is a still image. The **frame rate** is the number of frames displayed per second, measured in frames per second (fps). ### 3.5.1 Digital Video **Digital video** is video that has been converted into a digital format. Each frame of video is essentially an image, and video data is a sequence of these images displayed in rapid succession. Common video formats include: * **MP4** (MPEG-4 Part 14): A widely used video format that can store video and audio data. * **AVI** (Audio Video Interleave): An older video format that is less commonly used today. * **MOV** (QuickTime File Format): A video format developed by Apple. * **MKV** (Matroska Video): A flexible video format that can store multiple video and audio streams, as well as subtitles and other metadata. ### 3.5.2 Video Properties Key properties of digital video include: * **Frame Rate:** The number of frames displayed per second (e.g., 24 fps, 30 fps, 60 fps). * **Resolution:** The width and height of each frame in pixels (e.g., 1920x1080 for Full HD). * **Bitrate:** The amount of data used to store each second of video (measured in bits per second or megabits per second). ### 3.5.3 Video Compression **Video compression** is the process of reducing the file size of a video file. There are two types of video compression: * **Lossy compression:** Reduces the file size by discarding some of the video data. This can result in a loss of video quality, but the file size is significantly reduced. * **Lossless compression:** Reduces the file size without discarding any of the video data. This results in a smaller file size than uncompressed video, but the video quality is the same as the original. Common video codecs (coder-decoders) include: * **H.264** (Advanced Video Coding): A widely used video codec that provides good compression and video quality. * **H.265** (High Efficiency Video Coding): A newer video codec that provides better compression than H.264, allowing for smaller file sizes and higher video quality. * **VP9:** An open-source video codec developed by Google. ### Summary This chapter discussed how different types of data are represented in computers. Numeric data is represented using the binary number system, with integers represented using two's complement and floating-point numbers represented using the IEEE 754 standard. Text data is represented using character encoding schemes such as ASCII and Unicode. Image data is represented as a grid of pixels in raster graphics or using mathematical equations in vector graphics. Audio data is represented as a sequence of samples, and video data is represented as a sequence of frames. Both audio and video data can be compressed using lossy or lossless compression techniques.